Oblivious tight compaction in O(n) time with smaller constant

Oblivious compaction is a crucial building block for hash-based oblivious RAM. Asharov et al. recently gave a O(n) algorithm for oblivious tight compaction. Their algorithm is deterministic and asymptotically optimal, but it is not practical to implement because the implied constant is $\gg 2^{38}$. We give a new algorithm for oblivious tight compaction that runs in time $< 16014.54n$. As part of our construction, we give a new result in the bootstrap percolation of random regular graphs

On some problems of Lovász concerning the shannon capacity of a graph

The answers to several problems of Lovász concerning the Shannon capacity of a graph are shown to be negative

The Hermitian two-graph and its code.

AMS classifications: 05E30; 51E20; 94B05;

The pseudo-geometric graphs for generalized quadrangles of order (3,t)

AbstractThe values t= 1, 3, 5, 6, 9 satisfy the standard necessary conditions for existence of a generalized quadrangle of order (3, t). This gives the following possible parameter sets for strongly regular graphs that are pseudo-geometric for such a generalized quadrangle: (v, k,λ , μ) = (16, 6, 2, 2), (40, 12, 2, 4), (64, 18, 2, 6), (76, 21, 2, 7) and (112, 30, 2, 10). It is well known that there are two graphs with the first parameter set and that there is just one graph with the last set of parameters. Recently, the second author has shown that there are precisely 28 strongly regular graphs with the second parameter set. Non-existence of a strongly regular graph with the fourth set of parameters was proved by the first author. Here we complete the classification by announcing that there are exactly 167 non-isomorphic strongly regular graphs with parameters (64, 18, 2, 6)

A partial geometry pg(9,8,4)

We describe a construction of a partial geometry pg(9, 8, 4) based on binary words of length 9 and PG (1, 8)

Spreads in strongly regular graphs

Dedicated to Hanfried Lenz on the occasion of his 80th birthday Abstract. A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte&apos;s bound (also called Hoffman&apos;s bound). Such spreads give rise to colorings meeting Hoffman&apos;s lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences l ad to conditions for existence. Most examples come from spreads and fans in (partial) geometries. We give other examples, including a spread in the McLaughlin graph. For strongly regular graphs related to regular two-graphs, spreads give lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph

Spreads in strongly regular graphs

A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte&apos;s bound (also called Hoffman&apos;s bound). Such spreads give rise to colorings meeting Hoffman&apos;s lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (partial) geometries. We give other examples, including a spread in the McLaughlin graph. For strongly regular graphs related to regular two-graphs, spreads give lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph

Cospectral graphs and the generalized adjacency matrix

Let J be the all-ones rnatrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ - A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ - A for exactly one value (y) over cap of (y) over cap. We call such graphs (y) over cap -cospectral. It follows that is a rational number, and we prove existence of a pair of (y) over cap -cospectral graphs for every rational. In addition, we generate by computer all (y) over cap -cospectral pairs on at most nine vertices. Recently, Chesnokov and the second author constructed pairs of (y) over cap -cospectral graphs for all rational (y) over cap is an element of (0, 1), where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of, and by computer we find all Such pairs of (y) over cap -cospectral graphs on at most eleven vertices. (C) 2006 Elsevier Inc. All rights reserved.X1116sciescopu